I. Research Background: Why Conduct a Study on School Stereotyping?

Currently, laser diffraction, sieving, and microscopic observation are the commonly used methods for measuring powder particle size. Among these, laser diffraction spectroscopy (LSD) offers high precision and is often regarded as the reference standard; however, the associated instrumentation is relatively expensive and requires specialized expertise. Meanwhile… Polymer encapsulation + optical microscopy observation Low cost and simple operation make it the mainstream choice for small and medium-sized manufacturers in the industry.

However, this method suffers from two inherent biases:

1. Sectional deviation : It is virtually impossible for the observation plane to pass through the region of the particle’s maximum diameter, so the dimensions measured from two-dimensional cross-sections are generally smaller than the particle’s true three-dimensional size.
2. Emergent bias : Large particles have a higher probability of being detected, which can lead to discrepancies between the two-dimensional particle size distribution and the true three-dimensional distribution.

To obtain an accurate particle size distribution from cross-sectional images, Stereoscopic school is correct. This has become the key. This study focuses on materials commonly used in additive manufacturing. Ti-6Al-4V titanium alloy powder For the target material, powders were prepared using three mainstream processes—plasma atomization, gas atomization, and ultrasonic atomization—and the practical effectiveness of three classic volumetric calibration methods was systematically evaluated.

II. Experimental Design: Materials, Procedures, and Three Correction Methods

1. Sample Preparation and Testing Procedure

The study employed three types of Ti‑6Al‑4V powders and carried out a complete set of sample preparation and analysis procedures:

• Embedding sample preparation : Embed the powder into the polymer, then grind and polish to produce a flat observation cross-section;
• Image Acquisition : Cross-sectional images captured using a 200× microscope; for single‑class powder analysis, 2,300–2,600 particle cross sections are examined to ensure data reliability.
• Image analysis : Image segmentation is performed using MATLAB algorithms, and the equivalent circle diameter of particle cross-sections is calculated to obtain the original apparent particle size distribution.
• Benchmark comparison : The results obtained by the industry‑standard laser diffraction method (LSD) are used as the reference standard for true particle size.

2. The Three Major Methods of Stereoscopic Schooling

This study compares three mainstream correction techniques currently used in the field; all methods assume that powder particles are spherical, which aligns with the actual morphological characteristics of additive manufacturing feedstock powders:

• SSS method (Scheil-Schwartz-Saltykov) : A classic stereological algorithm that derives the correspondence between cross-sectional measurements and true particle sizes using probabilistic formulas; it has a long history of application.
• GCO Law (Goldsmith-Cruz-Orive) : Optimizing the traditional algorithm by calculating the distribution using the median of each particle-size interval is an industry‑recommended improved approach.
• FDM method (Finite Difference Method) : A correction algorithm based on integral equations and probabilistic models, which is also the method most strongly recommended in this study.

The study employed Mean Absolute Error (MAE) When evaluating the correction performance, a lower error value indicates that the corrected particle size distribution is closer to the true distribution.

III. Core Experimental Results: Comparative Analysis of Three Powders and Three Methods

The three types of powders prepared by different processes exhibit distinct particle-size characteristics: plasma-atomized and gas-atomized powders have finer particles with a log‑normal particle-size distribution, whereas ultrasonic‑treated powder shows a broader particle size range and a bimodal distribution. For all samples, the apparent particle sizes measured on the original cross‑section are significantly smaller than the LSD‑measured true particle sizes, further underscoring the necessity of correction.

1. Plasma-atomized powder

The original cumulative error in apparent particle size was 5.3%; after correction, the error decreased across the board:

• SSS Method: Cumulative Error 3.5%
• GCO method: cumulative error 3.0%
• FDM method: cumulative error is only 2.4% (Minimum error)

Whether considering cumulative error or piecewise differential error, the finite difference method (FDM) delivers the best fitting performance.

2. Gas-atomized powder

The cumulative error in the original apparent particle size is 3.8%, indicating a relatively small inherent bias:

• SSS Method: Cumulative error 3.7% (slight improvement)
• GCO method: cumulative error 2.8% (Overall best fit)
• FDM method: cumulative error 3.1%

In this powder, the GCO method demonstrates overall superior performance, while the FDM method exhibits a slight advantage in the differential error of segmented particle sizes.

3. Ultrasonic Treatment of Powders

For this type of powder, the initial measurement bias is the largest, with a cumulative error in apparent particle size reaching as high as 11.9%, making correction the most urgently needed:

• SSS method: cumulative error 8.7%
• GCO method: cumulative error 8.1%
• FDM method: cumulative error 7.5% (Minimum error)

All three methods significantly reduce detection bias, and FDM remains the approach with the best overall estimation performance.

4. Overall Data Trends

1. All correction methods can effectively address the underestimation of particle size caused by the sampling cross-section; after correction, the mean particle size falls between the apparent value and the true value.
2. The particle size distributions of all three types of powders are as follows: Right-skewed , the proportion of fine particles is higher than that of coarse particles; plasma- and gas‑atomized powders exhibit a sharp‑peaked size distribution, whereas ultrasonic‑atomized powders tend to show a more flat‑peaked distribution;
3. Compared with the traditional SSS method, the two novel algorithms—GCO and FDM—exhibit superior overall accuracy.

IV. Research Conclusions and Recommendations for Industry Applications

1. Core Conclusion

• Particle size distribution obtained solely from cross-sectional observations via polymer embedding There is a clear undervaluation. , stereoscopic inspection is an indispensable step in low-cost testing processes;
• All three correction methods can improve the accuracy of particle size estimation. Based on the test results for the three types of powders: The finite difference method (FDM) offers the best overall performance. , with the highest compatibility;
• Segmented scenario reference: Gas atomization powders are best suited to the GCO process; plasma atomization and ultrasonic atomization powders are ideally processed using FDM.
• Even after calibration, the ultrasonic‑processed powder still exhibits a 7.5% error; further algorithmic optimization is required for powders with large particle sizes and bimodal size distributions.

2. Practical Industry Recommendations

• For small and medium-sized additive manufacturing enterprises that lack high-end laser particle size analyzers, they may continue to use… Polymer encapsulation + microscopic observation + stereoscopic correction A combined solution that enables low-cost routine quality control of powders.
• Algorithm selection: For conventional titanium alloy spherical powders, the fused deposition modeling (FDM) process is preferred; if fine gas‑atomized powders are used as the primary feedstock, the GCO method may be chosen.
• Sample preparation requirements: For each batch, it is recommended to analyze more than 1,000 particle cross-sections to minimize statistical error and ensure the stability of calibration results.

V. Summary

Powder particle size is a core parameter in the quality control of additive manufacturing feedstock, and this study has filled a critical gap. Low-cost microscopic detection the limitation in accuracy. By combining conventional cross-sectional measurements with an appropriate volumetric stereoscopic correction algorithm, it is possible to both contain inspection costs and ensure data accuracy. Considering overall precision and versatility, Finite Difference Method (FDM) It has become the optimal solution for powders compatible with a wide range of atomization processes, while also providing a standardized reference framework for the entire additive manufacturing powder‑inspection field.

Image and Text Guide

Figure 1. Scanning electron microscope (SEM) images of Ti-6Al-4V powders prepared by (a) plasma atomization, (b) gas atomization, and (c) ultrasonic treatment.

Figure 2. Digital images of cross-sections of Ti‑6Al‑4V powder particles prepared by (a) plasma atomization, (b) gas atomization, and (c) ultrasonic treatment.

Figure 3. Cross-sectional images of Ti‑6Al‑4V powder particles prepared by (a) plasma atomization, (b) gas atomization, and (c) ultrasonic treatment.

Figure 4. The histograms show the particle size distributions obtained from stereological correction results based on apparent cross-sectional dimensions (Na) and the SSS, GCO, and FDM methods, corresponding respectively to (a) plasma‑atomized powder, (b) gas‑atomized powder, and (c) ultrasonic‑treated powder.

Figure 5. The cumulative line graph compares the particle size distributions obtained from stereological correction results based on apparent cross-sectional dimensions (Na), laser diffraction (LSD), and the SSS, GCO, and FDM methods, for (a) plasma‑atomized powder, (b) gas‑atomized powder, and (c) ultrasonically treated powder.

Table 1 presents the statistical measures of central tendency and dispersion for each particle-size distribution, along with the mean absolute error.